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Theorem oaordi 7671
Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
Assertion
Ref Expression
oaordi ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Proof of Theorem oaordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 5786 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21adantll 750 . . . 4 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
3 eloni 5771 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
4 ordsucss 7060 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
53, 4syl 17 . . . . . . . 8 (𝐵 ∈ On → (𝐴𝐵 → suc 𝐴𝐵))
65ad2antlr 763 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → suc 𝐴𝐵))
7 sucelon 7059 . . . . . . . . . 10 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
8 oveq2 6698 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝐴))
98sseq2d 3666 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)))
109imbi2d 329 . . . . . . . . . . . 12 (𝑥 = suc 𝐴 → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴))))
11 oveq2 6698 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝑦))
1211sseq2d 3666 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦)))
1312imbi2d 329 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦))))
14 oveq2 6698 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝑦))
1514sseq2d 3666 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
1615imbi2d 329 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
17 oveq2 6698 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝐵))
1817sseq2d 3666 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
1918imbi2d 329 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
20 ssid 3657 . . . . . . . . . . . . 13 (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)
21202a1i 12 . . . . . . . . . . . 12 (suc 𝐴 ∈ On → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)))
22 sssucid 5840 . . . . . . . . . . . . . . . . 17 (𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦)
23 sstr2 3643 . . . . . . . . . . . . . . . . 17 ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → ((𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦)))
2422, 23mpi 20 . . . . . . . . . . . . . . . 16 ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦))
25 oasuc 7649 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2625ancoms 468 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2726sseq2d 3666 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦)))
2824, 27syl5ibr 236 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
2928ex 449 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3029ad2antrr 762 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑦) → (𝐶 ∈ On → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3130a2d 29 . . . . . . . . . . . 12 (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑦) → ((𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦)) → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
32 sucssel 5857 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
337, 32sylbir 225 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On → (suc 𝐴𝑥𝐴𝑥))
34 limsuc 7091 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑥 → (𝐴𝑥 ↔ suc 𝐴𝑥))
3534biimpd 219 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑥 → (𝐴𝑥 → suc 𝐴𝑥))
3633, 35sylan9r 691 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴𝑥 → suc 𝐴𝑥))
3736imp 444 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → suc 𝐴𝑥)
38 oveq2 6698 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝐴 → (𝐶 +𝑜 𝑦) = (𝐶 +𝑜 suc 𝐴))
3938ssiun2s 4596 . . . . . . . . . . . . . . . . 17 (suc 𝐴𝑥 → (𝐶 +𝑜 suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +𝑜 𝑦))
4037, 39syl 17 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → (𝐶 +𝑜 suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +𝑜 𝑦))
4140adantr 480 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 suc 𝐴) ⊆ 𝑦𝑥 (𝐶 +𝑜 𝑦))
42 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
43 oalim 7657 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4442, 43mpanr1 719 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4544ancoms 468 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥𝐶 ∈ On) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4645adantlr 751 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4746adantlr 751 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 𝑥) = 𝑦𝑥 (𝐶 +𝑜 𝑦))
4841, 47sseqtr4d 3675 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) ∧ 𝐶 ∈ On) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥))
4948ex 449 . . . . . . . . . . . . 13 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)))
5049a1d 25 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝑥) → (∀𝑦𝑥 (suc 𝐴𝑦 → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦))) → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥))))
5110, 13, 16, 19, 21, 31, 50tfindsg 7102 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴𝐵) → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
5251exp31 629 . . . . . . . . . 10 (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
537, 52syl5bi 232 . . . . . . . . 9 (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 ∈ On → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
5453com4r 94 . . . . . . . 8 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
5554imp31 447 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
56 oasuc 7649 . . . . . . . . . 10 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +𝑜 suc 𝐴) = suc (𝐶 +𝑜 𝐴))
5756sseq1d 3665 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) ↔ suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58 ovex 6718 . . . . . . . . . 10 (𝐶 +𝑜 𝐴) ∈ V
59 sucssel 5857 . . . . . . . . . 10 ((𝐶 +𝑜 𝐴) ∈ V → (suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6058, 59ax-mp 5 . . . . . . . . 9 (suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
6157, 60syl6bi 243 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6261adantlr 751 . . . . . . 7 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
636, 55, 623syld 60 . . . . . 6 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6463imp 444 . . . . 5 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
6564an32s 863 . . . 4 ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) ∧ 𝐴 ∈ On) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
662, 65mpdan 703 . . 3 (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
6766ex 449 . 2 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
6867ancoms 468 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607   ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609
This theorem is referenced by:  oaord  7672  oaass  7686  odi  7704
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